Research

Research

My research area of interest is numerical analysis and applied mathematics. Key words describing my research interests are:

  • Numerical Partial Differential Equations (PDEs)
  • Finite Element Methods (FEMs)
  • Continuous Data Assimilation
  • Multigrid Methods
  • Phase-field Models
  • Cahn-Hilliard Equation and Cahn-Hilliard Coupled Systmes
  • Phase Field Crystal Equations
  • Other High-Order PDEs
  • Fluid Dynamics
  • Liquid Crystals

Details on specific projects can be found below.

Awards

NSF - DMS 2110768 (PI)

NSF - DMS 1952848 (Co-PI)

NSF - DMR 1950208 (Senior Personnel)

 

Continuous Data Assimilation

Continuous data assimilation numerical methods allow for data at only a few observable locations in space to be brought into a numerical method continuously over time to correct for defects in the initial conditions assigned at the start of the numerical method.

Sixth-Order Partial Differential Equations

Two relatively recent projects involving sixth-order (in space) partial differential equations include phase field crystal (PFC) models and microemulsions (ME) systems. The importance of PFC models stems from the fact that these models are able to increase accuracy (in comparison to lower order systems) in capturing the periodic nature of crystalline phases inherent in many different materials. ME systems are growing in popularity due to their ability in capturing many essential static properties of ternary oil-water-surfactant systems.

Solvers

Preconditioned MINRES

A preconditioned minimal residual (MINRES) algorithm with a block diagonal, symmetric, positive definite preconditioner was developed for first- and second-order in time finite element methods for the Cahn-Hilliard equation. The solver performance is independent of time and space step sizes, given any interfacial width parameter. Moreover, due to the block diagonal nature of the preconditioner, a multigrid method may be employed creating large computational savings.

Preconditioned Steepest Descent Solver

A preconditioned steepest descent (PSD) solver was developed for a finite difference method for the Cahn-Hilliard equation with a logarithmic Florry-Huggins energy potential. A convex splitting strategy is applied for the temporal discretization. However, the nonlinear and singular nature of the logarithmic energy potential posses many challenges to the numerical implementation. Fortunately, our PSD solver ensures a positivity-preserving property at each iteration stage thereby allowing efficient and accurate numerical experiments.

Phase-field Models

A large part of my research has been focused on phase-field models utilizing the Cahn-Hilliard equation. Specifically, I have focused on the development of stable and convergent finite element methods to solve the Cahn-Hilliard equation and Cahn-Hilliard coupled systems. The coupled systems included:

  • Cahn-Hilliard Darcy-Stokes
    • models the flow of a very viscous block copolymer fluid
  • Cahn-Hilliard Navier-Stokes
    • models two-phase fluid flow
  • Cahn-Hilliard Liquid Crystal
    • models liquid crystal droplets in a pure liquid crystal substance

I have also focused on developing solvers to improve on the efficiency of the finite element methods. This is described in the previous section.